A uniform magnetic field \(\mathbf{B}_{0}\) has only a \(z\)-component. Show that a vector potential given by \(A_{x}=-\frac{1}{2} B_{0} y, A_{y}=1, B_{0} x, A_{n}=0\) is suitable, but that so is \(A_{x}=-B_{0} y, A,=A_{z}=0 .\) Explain.

The magnetic field is a fundamental concept in physics, representing the vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A magnetic field has both a direction and a magnitude, and it is often visualized by drawing lines of force, or field lines, that indicate the direction of the field.

For example, in the given exercise, the magnetic field \( \mathbf{B}_{0} \) has only a z-component, which implies that it acts along the z-axis. This field configuration is quite common in physics problems as it simplifies calculations and can represent real-life situations like the uniform field inside a long, straight solenoid.

Knowing the vector representation of a magnetic field is crucial to solving many problems in electromagnetism, including finding vector potentials that result in the given field when applying operations like the curl.

In vector calculus, the curl operation measures the rotation of a vector field. In the context of electromagnetism, it has particular significance as it relates to Maxwell's equations, which govern how electric and magnetic fields are generated and altered. Specifically, the curl of a magnetic vector potential gives us the magnetic field.

In the exercise, we use the curl to verify whether a given vector potential correctly describes a uniform magnetic field. The curl is computed as the cross product of the gradient operator \( abla \) and the vector potential \( \vec{A} \) and is expressed in terms of the determinant involving partial derivatives. The resulting vector, after the curl is calculated, is compared to the known magnetic field to confirm the validity of the potential.

This step-by-step verification process shows us how vector potentials can be used to create specific magnetic field configurations and is essential for understanding the relationship between a potential and its resulting field.

Determinants play a key role in vector calculus, specifically when computing vector operations such as the curl. In the provided solutions, determinants are used to calculate the curl of vector potentials. They allow us to perform cross products in a structured manner by setting up a matrix with unit vectors, partial derivatives, and the vector field components.

The determinant of this matrix represents the curled vector in three-dimensional space. Calculating a determinant for a 3x3 matrix involves eliminating one row and one column for each element of the first row (the unit vectors), and finding the resulting 2x2 determinants for each. These values are then combined with the correct sign to yield the components of the curled vector.

Understanding how to use determinants in this context is essential for physics and engineering students as it enables them to solve complex problems involving magnetic fields and their potentials with clarity and precision.